Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Real-Time Adaptive Safety-Critical Control with Gaussian Processes in High-Order Uncertain Models

Yu Zhang, Long Wen, Xiangtong Yao, Zhenshan Bing, Linghuan Kong, Wei He, Alois Knoll

TL;DR

This paper presents an adaptive online learning framework for systems with uncertain parameters to ensure safety-critical control in non-stationary environments and proposes a safety filter based on high order control barrier functions (HOCBFs), synergized with the previously trained learning model.

Abstract

This paper presents an adaptive online learning framework for systems with uncertain parameters to ensure safety-critical control in non-stationary environments. Our approach consists of two phases. The initial phase is centered on a novel sparse Gaussian process (GP) framework. We first integrate a forgetting factor to refine a variational sparse GP algorithm, thus enhancing its adaptability. Subsequently, the hyperparameters of the Gaussian model are trained with a specially compound kernel, and the Gaussian model's online inferential capability and computational efficiency are strengthened by updating a solitary inducing point derived from new samples, in conjunction with the learned hyperparameters. In the second phase, we propose a safety filter based on high-order control barrier functions (HOCBFs), synergized with the previously trained learning model. By leveraging the compound kernel from the first phase, we effectively address the inherent limitations of GPs in handling high-dimensional problems for real-time applications. The derived controller ensures a rigorous lower bound on the probability of satisfying the safety specification. Finally, the efficacy of our proposed algorithm is demonstrated through real-time obstacle avoidance experiments executed using both a simulation platform and a real-world 7-DOF robot.

Real-Time Adaptive Safety-Critical Control with Gaussian Processes in High-Order Uncertain Models

TL;DR

This paper presents an adaptive online learning framework for systems with uncertain parameters to ensure safety-critical control in non-stationary environments and proposes a safety filter based on high order control barrier functions (HOCBFs), synergized with the previously trained learning model.

Abstract

This paper presents an adaptive online learning framework for systems with uncertain parameters to ensure safety-critical control in non-stationary environments. Our approach consists of two phases. The initial phase is centered on a novel sparse Gaussian process (GP) framework. We first integrate a forgetting factor to refine a variational sparse GP algorithm, thus enhancing its adaptability. Subsequently, the hyperparameters of the Gaussian model are trained with a specially compound kernel, and the Gaussian model's online inferential capability and computational efficiency are strengthened by updating a solitary inducing point derived from new samples, in conjunction with the learned hyperparameters. In the second phase, we propose a safety filter based on high-order control barrier functions (HOCBFs), synergized with the previously trained learning model. By leveraging the compound kernel from the first phase, we effectively address the inherent limitations of GPs in handling high-dimensional problems for real-time applications. The derived controller ensures a rigorous lower bound on the probability of satisfying the safety specification. Finally, the efficacy of our proposed algorithm is demonstrated through real-time obstacle avoidance experiments executed using both a simulation platform and a real-world 7-DOF robot.
Paper Structure (10 sections, 2 theorems, 30 equations, 6 figures, 1 table, 1 algorithm)

This paper contains 10 sections, 2 theorems, 30 equations, 6 figures, 1 table, 1 algorithm.

Table of Contents

  1. INTRODUCTION
  2. Background & Preliminaries
  3. High Order Control Barrier Function
  4. Variational Sparse Gaussian Processes
  5. Adaptive Online VSGP with the Affine Dot Product Kernel
  6. Offline Training for Hyperparameter
  7. Adaptive Online Learning for Data and Inducing Points
  8. Proposed Safety Controller
  9. Simulation and Real-World Experiments
  10. Conclusion

Key Result

Proposition 1

(Sufficient condition for safety) Consider the zero super-level set of a continuously differentiable control barrier function $h(\boldsymbol{x})$, denoted as $\mathcal{F}$. If the condition $x_{0} \in C_1 \cap \cdots \cap C_m$ holds, then the set $C_1 \cap \cdots \cap C_m$ is forward invariant, and

Figures (6)

  • Figure 1: An overview of the proposed AFVSGP-HOCBF framework. AFVSGP estimates the model-induced uncertainty affecting HOCBF, while HOCBF is applied to guarantee system safety.
  • Figure 2: Comparative analysis of predictive distributions across various learning methods under Gazebo simulation platform. Blue lines indicate the discrepancies between nominal estimates and actual model values, while red lines represent predictions from different models. A higher degree of alignment between red and blue lines suggests superior model performance. TUC represents the uncertain component with HOCBF and the blue shaded region represents the 95% confidence interval (the same applies to other figures).
  • Figure 3: Methods are initially trained and assessed using the end-effector's data on the y-axis. Subsequently, the motion is shifted to the z-axis to evaluate adaptability with previously unseen data.
  • Figure 4: Control barrier function across different learning methods during dynamic obstacle avoidance. Here, $h(x)$ represents the minimum distance to the unsafe area's boundary.
  • Figure 5: Online learning outcomes for real-world obstacle avoidance with the Franka robot using the AFVSGP-HOCBF method.
  • ...and 1 more figures

Theorems & Definitions (6)

  • Definition 1
  • Definition 2
  • Definition 3
  • Proposition 1
  • Definition 4
  • Proposition 2